#502020

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33610	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

515 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,24)(12,19)(13,35)(14,40)(15,22)(16,44)(17,23)(18,27)(20,26)(21,36)(25,57)(28,59)(29,61)(30,55)(31,48)(32,58)(33,53)(34,54)(37,46)(38,70)(39,51)(41,45)(42,60)(43,47)(49,62)(50,66)(52,67)(56,65)(63,72)(64,69)(68,71);(1,3,4,7)(2,5,6,8)(9,11,25,62,16,28,14,27,17,29,36,31,15,42,41,66)(10,18,45,59,22,49,21,24,23,50,40,60,44,48,57,61)(12,30,63,38,33,26,32,39,34,64,67,37,43,65,68,13)(19,51,71,20,47,70,52,55,54,35,58,56,53,46,72,69);(2,6)(5,8)(9,12)(10,19)(11,26)(13,31)(14,32)(15,43)(16,33)(17,34)(18,46)(20,24)(21,52)(22,47)(23,54)(25,63)(27,37)(28,64)(29,65)(30,42)(35,48)(36,67)(38,66)(39,62)(40,58)(41,68)(44,53)(45,71)(49,51)(50,70)(55,60)(56,61)(57,72)(59,69);(9,13,36,64,16,38,41,65,17,39,25,30,15,37,14,26)(10,20,40,46,22,55,57,51,23,56,45,70,44,69,21,35)(11,12,31,67,28,33,66,68,29,34,62,63,42,43,27,32)(18,47,60,72,49,54,61,71,50,53,59,52,48,19,24,58)

Polycyclic presentation:

2^9|b^2=efhi,d^2=eh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409216);